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# Population parameter estimation using SAEM

### Purpose

The estimation of the population parameters is performed using SAEM algorithm. The SAEM algorithm is the stochastic approximation expectation-maximization algorithm. It has been shown to be extremely efficient for a wide variety of complex models: categorical data, count data, time-to-event data, mixture models, differential equation based models, censored data, … Furthermore, convergence of SAEM has been rigorously proved and the implementation in Monolix is efficient. This is the only algorithm proposed in Monolix. We describe in the following some elements and settings of the algorithm.

### Population parameter estimation iterations

SAEM is an iterative algorithm: SAEM as implemented in Monolix has two phases. The goal of the first is to get to a neighborhood of the solution in only a few iterations. A simulated annealing version of SAEM accelerates this process when the initial value is far from the actual solution. The second phase consists of convergence to the located maximum with behavior that becomes increasingly deterministic, like a gradient algorithm. The sequence of estimated parameters ($\theta_k$) with respect to the iterations is displayed.

The two phases can be clearly seen on the figure above. Before the red line, the algorithm is exploring the possible solutions, while the second phase is looking for the minimum.

### Population parameter estimates

Also, the final estimations are displayed in the Results frame.

Notice that we split the results

• The fixed effects parameters (including “beta”-parameters, ti;e. the parameters associating the covariates)
• The associated standard deviation (if any). In case of IOV, the associated standard error of each level will be proposed in this part.
• The parameters from the error model (if any)

In terms of output, the population parameter estimation is proposed in a file populationParameters.txt (in the result folder) where the table of parameter name and value is defined.

### Main population parameter settings

The settings are accessible through the interface by the button next to the parameter estimation task.

During the burn-in phase: The user can choose the number of iteration in this phase before the exploration. It corresponds to an “initialization” of SAEM. The default value is 5.

During the exploratory phase: In this phase, the user can define

• the stopping rule, i.e. is the auto-stop criteria is used or the number of iteration is fixed (the auto-stop option is used by default),
• the maximum number of iteration (500 by default),
• the minimum number of iteration (in case of auto-stop use), i.e. corresponding to the interval length for the auto-stop criteria checks  (150 by default),
• the step-size exponent, corresponding to the memory for each iteration (0 by default)

Simulated annealing: A Simulated Annealing version of SAEM is used to estimate the population parameters (the variances are constrained to decrease slowly during the first iterations of SAEM).

In case of parameter without variability, an additional option is proposed between several methods.

During the smoothing phase: In this phase, the user can define

• the stopping rule, i.e. is the auto-stop criteria is used or the number of iteration is fixed (the auto-stop option is used by default),
• the maximum number of iteration (200 by default),
• the minimum number of iteration (in case of auto-stop use), i.e. corresponding to the interval length for the auto-stop criteria checks  (50 by default),
• the step-size exponent, corresponding to the memory for each iteration (1 by default)

### What is and what is not SAEM? And the related implications.

The objective of this paragraph is to explain what is the SAEM, what the user can wait for and its limitation.

SAEM is a stochastic algorithm: We cannot claim that SAEM always converges (i.e., with probability 1) to the global maximum of the likelihood. We can only say that it converges under quite general hypotheses to a maximum – global or perhaps local – of the likelihood. A large number of simulation studies have shown that SAEM converges with high probability to a “good” solution – it is hoped the global maximum – after a small number of iterations. Therefore, we encourage the user to use the convergence assessment tool to look at the sensitivity of the results w.r.t. the initial parameters

SAEM is a stochastic algorithm (2)The trajectory of the outputs of SAEM depends on the sequence of random numbers used by the algorithm. This sequence is entirely determined by the “seed.” In this way, two runs of SAEM using the same seed will produce exactly the same results. If different seeds are used, the trajectories will be different but convergence occurs to the same solution under quite general hypotheses. However, if the trajectories converge to different solutions, that does not mean that any of these results are “false”. It just means that the model is case sensitive to the seed or to the initial conditions.